Go to my iPad (or any iDevice): Settings: General: Reset: Erase All Content and Settings. This will wipe clean the iDevice, prompting you to set up as a new device or restore from an iTunes or iCloud backup.

Restore from iCloud backup. I had just deleted the note, so I chose an iCloud backup from the day before. You will need to be connected to your wifi network at this point. Let it go through the motions of restoring the information. It will give you an "estimated time remaining" status bar. You need to be by your router at this point. Once it finishes restoring, your iDevice will reboot. During the reboot, your iDevice will be loading all of the information that it just downloaded from the iCloud backup. At this point, your old note will be in there, but as soon as it loads, if you're connected to the internet, it will re-sync with iCloud and you'll will right back where you were before you did the restore. I know because I did it once already. To stop this from happening:

As soon as your iDevice starts to reboot, turn off your wifi router or otherwise make it to where the iDevice has no internet connection. Your iDevice will finish loading the iCloud backup, and your notes will be as they were the day before or at the time of the last back up.

I only had one note that I needed to retrieve, so I copied it and opened an email and emailed it to myself. I was a little concerned that this may not work, seeing as I was not connected to the internet, but, once I reconnected, the email sent. Naturally, at that point my notes were all synced back to current, but I had the note in the email and, thus, was able to recreate the note.

Note: If you have a cellular device, pull out the sim before you start this process. If not it will still sync the new notes even with the WiFi off.

Logging in just to say that Michiel's answer worked for me as I just had this issue and was able to retrieve the mistakenly deleted note by doing as he describes, and as apparently I need 15 rep to up-vote him so this is my only way to tell others it works.
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honestduaneApr 2 '13 at 23:54

Depends if you have any other devices that also sync. If so, then if you can stop them from syncing, you can at least copy the contents before it is removed. Other than that, once it's gone, it's gone. Their may be more potential for getting it back in Mountain Lion which has a dedicated Notes client and also the standard HFS filesystem which may allow you to "undelete", but this is conjecture at this point.